Tutorial on Non-Deterministic Semantics Part I:...

Preliminaries Non-deterministic Matrices: Basic Defs and Props Canonical Gentzen-type systems

Tutorial on Non-Deterministic SemanticsPart I: Introduction

Arnon Avron and Anna Zamansky

UNILOG 2013, Rio de Janeiro

What this tutorial is about

Non-deterministic Matrices:

Incorporating the notion of “non-deterministic computations” fromautomata and computability theory into logical truth-tables.

We would like to show:

Non-deterministic semantics is a natural and useful paradigm.

What this tutorial is about

Non-deterministic Matrices:

Incorporating the notion of “non-deterministic computations” fromautomata and computability theory into logical truth-tables.

We would like to show:

Non-deterministic semantics is a natural and useful paradigm.

Let’s start with a simple question

What are the roots of the equation x2 = 1?

Answer:

1 and -1

What are the roots of the equation x2 = −1?

Naive answer:

there are no ordinary (real) numbers...

Not satisfied?

introduce imaginary numbers!

Answer:

1 and -1

Naive answer:

Not satisfied?

Answer:

1 and -1

Naive answer:

Not satisfied?

Answer:

1 and -1

Naive answer:

Not satisfied?

Answer:

1 and -1

Naive answer:

Not satisfied?

Another simple question: what is the semantics of LK?

Γ⇒ ∆, ψ

Γ,¬ψ ⇒ ∆

Γ, ψ ⇒ ∆

Γ⇒ ∆,¬ψ

Γ, ψ ⇒ ∆ Γ, ϕ⇒ ∆

Γ, ψ ∨ ϕ⇒ ∆

Γ⇒ ∆, ψ, ϕ

Γ⇒ ∆, ψ ∨ ϕ

¬t ff t

∨t t tt f tf t tf f f

Another simple question: what is the semantics of LK?

Γ⇒ ∆, ψ

Γ,¬ψ ⇒ ∆

Γ, ψ ⇒ ∆

Γ⇒ ∆,¬ψ

Γ, ψ ⇒ ∆ Γ, ϕ⇒ ∆

Γ, ψ ∨ ϕ⇒ ∆

Γ⇒ ∆, ψ, ϕ

Γ⇒ ∆, ψ ∨ ϕ

¬t ff t

And of this system?

Γ⇒ ∆, ψ

Γ,¬ψ ⇒ ∆

Γ, ψ ⇒ ∆

Γ⇒ ∆,¬ψ

Γ, ψ ⇒ ∆ Γ, ϕ⇒ ∆

Γ, ψ ∨ ϕ⇒ ∆

Γ⇒ ∆, ψ, ϕ

Γ⇒ ∆, ψ ∨ ϕ

¬t ff ?

Theorem:

The above system has no finite-valued characteristic matrix.

And of this system?

Γ⇒ ∆, ψ

Γ,¬ψ ⇒ ∆

Γ, ψ ⇒ ∆

Γ⇒ ∆,¬ψ

Γ, ψ ⇒ ∆ Γ, ϕ⇒ ∆

Γ, ψ ∨ ϕ⇒ ∆

Γ⇒ ∆, ψ, ϕ

Γ⇒ ∆, ψ ∨ ϕ

¬t ff ?

Theorem:

The above system has no finite-valued characteristic matrix.

Not satisfied? Introduce non-deterministic matrices!

Γ⇒ ∆, ψ

Γ,¬ψ ⇒ ∆

Γ, ψ ⇒ ∆

Γ⇒ ∆,¬ψ

Γ, ψ ⇒ ∆ Γ, ϕ⇒ ∆

Γ, ψ ∨ ϕ⇒ ∆

Γ⇒ ∆, ψ, ϕ

Γ⇒ ∆, ψ ∨ ϕ

¬t ff {t,f}

The idea of non-determinististic semantics

Shutte (1960) and Girard (1987)

Ivlev (1998)

Batens (1998)

Crawford and Etherington (1998)

Avron and Lev (2001)

What is the non-deterministic paradigm useful for?

Modular semantic characterization of non-classical logics(Logics of Formal (In)consistency)

Systematic construction of cut-free sequent systems

Semantic tools for investigation of proof systems

Overview of Part I - Introduction

1 Preliminaries

2 Non-deterministic Matrices: Basic Defs and Props

3 Canonical Gentzen-type systems

Linguistic ambiguity

In many natural languages, “or” has both inclusive and exclusivemeanings.

If a mathematician says:

I shall either attack problem A or attack problem B

He may solve both problems, but in some situations he actuallymeans “...but don’t expect me to solve both”.

Capturing both meanings:

a b a OR b

t t {t, f}t f {t}f t {t}f f {f}

Linguistic ambiguity

In many natural languages, “or” has both inclusive and exclusivemeanings.

If a mathematician says:

I shall either attack problem A or attack problem B

He may solve both problems, but in some situations he actuallymeans “...but don’t expect me to solve both”.

Capturing both meanings:

a b a OR b

t t {t, f}t f {t}f t {t}f f {f}

Electrical circuits

Problems in the manufacturing process

Disturbing noise sources, temperature, etc.

Adversary operations

? t ft {t, f} {t}f {t} {f}

Evaluation with unknown computation models

Send A ∨ B for evaluation to a distant computer C

C performs either a parallel or a sequential computation onA ∨ B.

∨ f I t

f f I tI I I tt t t t

∨ f I t

f f I tI I I I

t t t t

Kleene McCarthy

∨ f I t

f {f} {I} {t}I {I} {I} {t, I}t {t} {t} {t}

How we will use non-deterministic truth tables

To characterize interesting logics which had so far no usefulsemantic characterizations.

Some important questions:

what is a logic?

how are logics characterized?

what characterizations are useful?

How we will use non-deterministic truth tables

To characterize interesting logics which had so far no usefulsemantic characterizations.

Some important questions:

what is a logic?

how are logics characterized?

what characterizations are useful?

What is a logic?

1 A formal language L.

2 A consequence relation ` for L.

consequence relation (cr) for La binary relation between sets of L-formulas and L-formulas with:

strong reflexivity: if ψ ∈ Γ then Γ ` ψ.monotonicity: if Γ ` ψ and Γ ⊆ Γ′, then Γ′ ` ψ.transitivity (cut): if Γ ` ψ and Γ, ψ ` ϕ then Γ ` ϕ.

Properties of CR ` for L

Structurality: for every uniform L-substitution σ and every Γand ψ: if Γ ` ψ then σ(Γ) ` σ(ψ).Example: p ∧ q ` q implies ϕ ∧ ψ ` ψ for every ϕ,ψ ∈ FL.

Consistency: there exist formulas ϕ and ψ, such that ϕ 6` ψ.

Finitariness: whenever Γ ` ψ, there exists some finite Γ′ ⊆ Γ,such that Γ′`ψ.

Propositional logic:

A pair 〈L,`〉, where ` is a structural, consistent and finitary cr.

How are logics characterized?

Semantically: Γ `S ψ if every “model” of Γ is a “model” of ψin the semantics S.

Syntactically: Γ `D ψ if ψ has a proof from Γ in the deductionsystem D.

Classical truth tables

¬t ff t

⊃ t ft t ff t t

∧ t ft t ff f f

∨ t ft t tf t f

A classical valuation:

any function v : Fcl → {t, f }, such that

v(�(ψ1, . . . , ψn)) = �(v(ψ1), . . . , v(ψn))

for any connective � ∈ {¬,⊃,∨,∧}.

Classical truth tables

¬t ff t

⊃ t ft t ff t t

∧ t ft t ff f f

∨ t ft t tf t f

A classical valuation:

any function v : Fcl → {t, f }, such that

v(�(ψ1, . . . , ψn)) = �(v(ψ1), . . . , v(ψn))

for any connective � ∈ {¬,⊃,∨,∧}.

Semantic way of defining classical logic

A classical valuation v is a model of an L-formula ψ ifv(ψ) = t. v is a model of a theory Γ if v is a model of everyψ ∈ Γ.

Γ `CPL ψ if every classical model of Γ is a model of ψ.

Hilbert-style proof systems

A Hilbert-style proof system for Lan (effective) set of axioms

an (effective) set of inference rules

A proof of ψ from Γ in H:

a finite sequence of L-formulas, where the last formula is ψ, andeach formula is:

an axiom of H, or

a member of Γ, or

is obtained from previous formulas in the sequence byapplying some inference rule of H.

Hilbert-style proof systems

A Hilbert-style proof system for Lan (effective) set of axioms

an (effective) set of inference rules

A proof of ψ from Γ in H:

a finite sequence of L-formulas, where the last formula is ψ, andeach formula is:

an axiom of H, or

a member of Γ, or

is obtained from previous formulas in the sequence byapplying some inference rule of H.

Axiom schemata:

I1 ϕ ⊃ (ψ ⊃ ϕ)I2 (ϕ ⊃ ψ ⊃ θ) ⊃ (ϕ ⊃ ψ) ⊃ (ϕ ⊃ θ)I3 ((ψ ⊃ ϕ) ⊃ ψ) ⊃ ψ

C1 ϕ ∧ ψ ⊃ ϕC2 ϕ ∧ ψ ⊃ ψC3 ϕ ⊃ (ψ ⊃ ϕ ∧ ψ)D1 ϕ ⊃ ϕ ∨ ψD2 ψ ⊃ ϕ ∨ ψD3 (ϕ ⊃ θ) ⊃ (ψ ⊃ θ) ⊃ (ϕ ∨ ψ ⊃ θ)N1 ¬ϕ ∨ ϕN2 (ϕ ∧ ¬ϕ) ⊃ ψ

Inference Rule:ψ ψ ⊃ ϕ

Soundness and completeness theorem for CPL:

Γ `HCL ψ ⇔ Γ `CPL ψ

Axiom schemata:

Soundness and completeness theorem for CPL:

Γ `HCL ψ ⇔ Γ `CPL ψ

Gentzen-style proof systems

Hilbert-style systems operate on L-formulas. Gentzen-stylesystems operate on sequents.

A sequent: an expression of the form Γ⇒ ∆, where Γ,∆ arefinite sets of L-formulas.

A standard Gentzen-type system for L:

1 Standard axioms: ψ ⇒ ψ.

2 Structural Weakening and Cut rules:

Γ⇒ ∆Γ, Γ′ ⇒ ∆,∆′

(Weakening)Γ, ψ ⇒ ∆ Γ⇒ ∆, ψ

Γ⇒ ∆(Cut)

3 Logical introduction rules for the connectives of L.

Gentzen-style proof systems

Hilbert-style systems operate on L-formulas. Gentzen-stylesystems operate on sequents.

A sequent: an expression of the form Γ⇒ ∆, where Γ,∆ arefinite sets of L-formulas.

A standard Gentzen-type system for L:

Γ⇒ ∆Γ, Γ′ ⇒ ∆,∆′

Γ⇒ ∆(Cut)

The system GCPL

ψ ⇒ ψ

(Weakening)Γ⇒ ∆

Γ, Γ′ ⇒ ∆,∆′ (Cut)Γ, ψ ⇒ ∆ Γ⇒ ∆, ψ

Γ⇒ ∆

(¬ ⇒)Γ⇒ ∆, ϕ

¬ϕ, Γ⇒ ∆(⇒ ¬)

ϕ, Γ⇒ ∆

Γ⇒ ∆,¬ϕ

(⊃⇒)Γ⇒ ∆, ϕ ψ, Γ⇒ ∆

ϕ ⊃ ψ, Γ⇒ ∆(⇒⊃)

Γ, ϕ⇒ ∆, ψ

Γ⇒ ∆, ϕ ⊃ ψ

(∧ ⇒)Γ, ϕ, ψ ⇒ ∆

Γ, ϕ ∧ ψ ⇒ ∆(⇒ ∧)

Γ⇒ ∆, ϕ Γ⇒ ∆, ψ

Γ⇒ ∆, ϕ ∧ ψ

(∨ ⇒)Γ, ϕ⇒ ∆ Γ, ψ ⇒ ∆

Γ, ϕ ∨ ψ ⇒ ∆(⇒ ∨)

Γ⇒ ∆, ϕ, ψ

Γ⇒ ∆, ϕ ∨ ψ

Completeness and cut-elimination

A Gentzen-style system admits cut-elimination if whenever Γ⇒ ∆is provable in G , Γ⇒ ∆ also has a cut-free proof in G .

Completeness Theorem for Classical Logic:

Γ `GCPL ψ iff Γ `CPL ψGCPL admits cut-elimination

Corollary: the subformula property

If Γ⇒ ∆ has a derivation in GCPL, then all the formulas in thisderivation are subformulas of the formulas in Γ⇒ ∆.

Basic principles of classical logic

Bivalence:Every proposition is either true or false (there are exactly twotruth-values).

Inconsistency Intolerance:A proposition and its negation cannot be both true.

Truth-Functionality:The truth-value of a complex proposition is uniquely definedby the truth-values of its constituents.

Many-valued logic - motivation

Sometimes incomplete information prevents us from telling ifsomething is true or not.

Lukasiewicz, “On Determinism”, 1922: If statements aboutfuture events are already true or false, then the future is asmuch determined as the past and differs from the past only inso far as it has not yet come to pass.

The idea: reject Bivalence by adding a third truth-value I, tobe read as “possible”.

Three-valued Lukasiewicz logic

⊃ f I t

f t t tI I t tt f I t

¬f tI It f

A legal valuation v is a Luk-model of a formula ψ if v(ψ) = t.v is a Luk-model of a theory Γ if v is a Luk-model of everyψ ∈ Γ.

Γ `Luk ψ if every Luk-model of Γ is a Luk-model of ψ.

Kleene and McCarthy logics

Modelling parallel vs. sequential computation

The third truth-value I - for “undefined”

Negation is defined like in Lukasiewicz three-valued logic.

The notion of a model of a formula and the associated cr aredefined like in Lukasiewicz three-valued logic.

∨ f I t

f f I tI I I tt t t t

∨ f I t

f f I tI I I It t t t

Kleene McCarthy

General semantic method for defining logics

A denotational semantics S = 〈S , |=S〉S - a non-empty set of “valuations” (usually mappings fromformulas to “truth values”)

|=S - a “satisfaction” relation between “valuations” andformulas

For v ∈ S , v is a S-model of ψ if v |=S ψ. v is an S-model ofΓ if v |=S ψ for every ψ ∈ Γ.

Γ `S ψ if every S-model of Γ is an S-model of ψ.

For any denotational semantics S = 〈S , |=S〉 for L, `S is a cr.

General semantic method for defining logics

A denotational semantics S = 〈S , |=S〉S - a non-empty set of “valuations” (usually mappings fromformulas to “truth values”)

|=S - a “satisfaction” relation between “valuations” andformulas

For v ∈ S , v is a S-model of ψ if v |=S ψ. v is an S-model ofΓ if v |=S ψ for every ψ ∈ Γ.

Γ `S ψ if every S-model of Γ is an S-model of ψ.

For any denotational semantics S = 〈S , |=S〉 for L, `S is a cr.

Many-valued matrices as denotational semantics

M = 〈V,D,O〉 is a matrix for a propositional language L if:

V is a nonempty set of truth-values,

∅ 6= D ⊂ V (the set of designated truth-values),

for every n-ary connective � of L, O includes an operation� : Vn → V

Examples

In classical logic: V = {t, f}, while D = {t}.

In the 3-valued logics of Lukasiewicz, Kleene, and McCarthy:V = {t, f, I}, while D = {t}.

Examples

In classical logic: V = {t, f}, while D = {t}.In the 3-valued logics of Lukasiewicz, Kleene, and McCarthy:V = {t, f, I}, while D = {t}.

Valuations

A valuation v in a matrix M = 〈V,D,O〉 is any function fromthe set of L-formulas to V such that:

v(�(ψ1, ..., ψn)) = �(v(ψ1), ..., v(ψn))

v is a model of an L-formula ψ in M, denoted by v |=M ψ, ifv(ψ) ∈ D.

v is a model of a theory Γ in M, denoted by v |=M Γ, if v is amodel of every ψ ∈ Γ.

Logics induced by matrices

Γ `M ψ if for every valuation v in M: v |=M Γ impliesv |=M ψ.

Let L = 〈L,`〉 be some logic.L is sound for a matrix M if ` ⊆ `M.L is complete for M if `M ⊆ `.M is a characteristic matrix for L if L is sound and completefor M.

For any matrix M for L, 〈L,`M〉 is a propositional logic.

Converse direction: is every propositional logic induced by amatrix?

Logics induced by matrices and their families

Logic induced by family C of matrices:

Γ `C ψ if for every M ∈ C : Γ `M ψ.

Every propositional logic is induced by some family ofmatrices.

A set of L-formulas (theory) Γ is `-consistent if there existssome L-formula ψ such that Γ6`ψ.

A logic L = 〈L,`〉 is uniform if for every two theories Γ1, Γ2

and an L-formula ψ: Γ1 ` ψ whenever Γ1, Γ2 ` ψ and Γ2 is a`-consistent theory with no atoms in common with Γ1 ∪ {ψ}.

Theorem ( Los & Suszko)

A finitary propositional logic has a characteristic matrix iff it isuniform.

Non-deterministic matrices

A non-deterministic matrix (Nmatrix) for L is a tupleM = 〈V,D,O〉:

V - the set of truth-values,

D - the set of designated truth-values,

O - contains an interpretation function � : Vn → P+(V) forevery n-ary connective � of L.

Ordinary matrices are a special case:

each � is a function taking singleton values only (can be treated asa function � : Vn → V).

Non-deterministic matrices

A non-deterministic matrix (Nmatrix) for L is a tupleM = 〈V,D,O〉:

V - the set of truth-values,

D - the set of designated truth-values,

O - contains an interpretation function � : Vn → P+(V) forevery n-ary connective � of L.

Ordinary matrices are a special case:

each � is a function taking singleton values only (can be treated asa function � : Vn → V).

Example 1: two truth values

L = {∨,∧,⊃,¬}

V = {f, t},D = {t}

∨,∧ and ⊃ are interpreted classically

¬ satisfies law of excluded middle (¬ϕ ∨ ϕ), but not law ofcontradiction (¬(ϕ ∧ ¬ϕ)).

The Nmatrix M2 = 〈V,D,O〉:

∨ ∧ ⊃t t {t} {t} {t}t f {t} {f} {f}f t {t} {f} {t}f f {f} {f} {t}

¬t {t, f}f {t}

Example 2: three truth values

Two 3-valued Nmatrices with V = {f, I, t}, D = {I, t}

M3L = 〈V,D,OL〉 M3

S = 〈V,D,OS〉

Standard interpretations of disjunction, conjunction andimplication:

a∨b =

{D if either a ∈ D or b ∈ D{f} if a = b = f

a∧b =

{D if a, b ∈ D{f} if either a = f or b = f

a⊃b =

{D if either a = f or b ∈ D{f} if a ∈ D and b = f

Negation:

¬t {f}I Vf {t}

¬t {f}I Df {t}

Non-deterministic matrices as denotational semantics

M = 〈V,D,O〉 - an Nmatrix for L.

An M-valuation v is any function from L-formulas to V whichsatisfies:

v(�(ψ1, ..., ψn)) ∈ �(v(ψ1), ..., v(ψn))

v |=M ψ if v(ψ) ∈ D.

v |=M Γ if v(ψ) ∈ D for every ψ ∈ Γ.

Consequence relation induced by a Nmatrix

Γ `M ψ if for every M-valuation v : v |=M Γ implies v |=M ψ.

v(�(ψ1, ..., ψn)) ∈ �(v(ψ1), ..., v(ψn))

The Nmatrix M2 revisited

¬t {t, f}f {t}

Any M2-valuation satisfies ¬ψ ∨ ψ but not necessarily ψ ⊃ ¬¬ψ:

v(p) = t v(¬p) = t v(¬¬p) = f

The logic generated by M2 is known as the basic adaptiveparaconsistent logic CLuN introduced by Batens.

The Nmatrix M2 revisited

¬t {t, f}f {t}

Any M2-valuation satisfies ¬ψ ∨ ψ but not necessarily ψ ⊃ ¬¬ψ:

v(p) = t v(¬p) = t v(¬¬p) = f

The logic generated by M2 is known as the basic adaptiveparaconsistent logic CLuN introduced by Batens.

Reminder: HCL

Axiom schemata:

Soundness and completeness theorem:

Γ `HCLuNψ ⇔ Γ `M2 ψ

Axiom schemata:

Γ `HCLuNψ ⇔ Γ `M2 ψ

The system GCPL

ψ ⇒ ψ

Γ, Γ′ ⇒ ∆,∆′ (Cut)Γ, ψ ⇒ ∆ Γ⇒ ∆, ψ

Γ⇒ ∆

(¬ ⇒)Γ⇒ ∆, ϕ¬ϕ, Γ⇒ ∆

(⇒ ¬)ϕ, Γ⇒ ∆

Γ⇒ ∆,¬ϕ

(⊃⇒)Γ⇒ ∆, ϕ ψ, Γ⇒ ∆

ϕ ⊃ ψ, Γ⇒ ∆(⇒⊃)

Γ, ϕ⇒ ∆, ψΓ⇒ ∆, ϕ ⊃ ψ

(∧ ⇒)Γ, ϕ, ψ ⇒ ∆

Γ, ϕ ∧ ψ ⇒ ∆(⇒ ∧)

Γ⇒ ∆, ϕ Γ⇒ ∆, ψΓ⇒ ∆, ϕ ∧ ψ

(∨ ⇒)Γ, ϕ⇒ ∆ Γ, ψ ⇒ ∆

Γ, ϕ ∨ ψ ⇒ ∆(⇒ ∨)

Γ⇒ ∆, ϕ, ψΓ⇒ ∆, ϕ ∨ ψ

The system GCLuN

ψ ⇒ ψ

Γ, Γ′ ⇒ ∆,∆′ (Cut)Γ, ψ ⇒ ∆ Γ⇒ ∆, ψ

Γ⇒ ∆

(¬ ⇒)Γ⇒ ∆, ϕ¬ϕ, Γ⇒ ∆

(⇒ ¬)ϕ, Γ⇒ ∆

Γ⇒ ∆,¬ϕ

(⊃⇒)Γ⇒ ∆, ϕ ψ, Γ⇒ ∆

ϕ ⊃ ψ, Γ⇒ ∆(⇒⊃)

Γ, ϕ⇒ ∆, ψΓ⇒ ∆, ϕ ⊃ ψ

(∧ ⇒)Γ, ϕ, ψ ⇒ ∆

Γ, ϕ ∧ ψ ⇒ ∆(⇒ ∧)

Γ⇒ ∆, ϕ Γ⇒ ∆, ψΓ⇒ ∆, ϕ ∧ ψ

(∨ ⇒)Γ, ϕ⇒ ∆ Γ, ψ ⇒ ∆

Γ, ϕ ∨ ψ ⇒ ∆(⇒ ∨)

Γ⇒ ∆, ϕ, ψΓ⇒ ∆, ϕ ∨ ψ

T `GCLuNψ ⇔ T `M2 ψ

Reminder: T `G ψ if there is some finite Γ ⊆ T , such that `G Γ⇒ ψ.

The system GCLuN

ψ ⇒ ψ

Γ, Γ′ ⇒ ∆,∆′ (Cut)Γ, ψ ⇒ ∆ Γ⇒ ∆, ψ

Γ⇒ ∆

(¬ ⇒)Γ⇒ ∆, ϕ¬ϕ, Γ⇒ ∆

(⇒ ¬)ϕ, Γ⇒ ∆

Γ⇒ ∆,¬ϕ

(⊃⇒)Γ⇒ ∆, ϕ ψ, Γ⇒ ∆

ϕ ⊃ ψ, Γ⇒ ∆(⇒⊃)

Γ, ϕ⇒ ∆, ψΓ⇒ ∆, ϕ ⊃ ψ

(∧ ⇒)Γ, ϕ, ψ ⇒ ∆

Γ, ϕ ∧ ψ ⇒ ∆(⇒ ∧)

Γ⇒ ∆, ϕ Γ⇒ ∆, ψΓ⇒ ∆, ϕ ∧ ψ

(∨ ⇒)Γ, ϕ⇒ ∆ Γ, ψ ⇒ ∆

Γ, ϕ ∨ ψ ⇒ ∆(⇒ ∨)

Γ⇒ ∆, ϕ, ψΓ⇒ ∆, ϕ ∨ ψ

T `GCLuNψ ⇔ T `M2 ψ

Reminder: T `G ψ if there is some finite Γ ⊆ T , such that `G Γ⇒ ψ.

M3L and M3

S revisited

V = {f, I, t}, D = {I, t}

M3L = 〈V,D,OL〉 M3

S = 〈V,D,OS〉

a∨b =

{D if either a ∈ D or b ∈ D{f} if a = b = f

a∧b =

{D if a, b ∈ D{f} if either a = f or b = f

a⊃b =

{D if either a = f or b ∈ D{f} if a ∈ D and b = f

¬t {f}I Vf {t}

¬t {f}I Df {t}

Proof system for M3L and M3

Let the Gentzen-style system GCmin be obtained by replacingthe rule (¬ ⇒) in GCPL with the rule:

Γ, ϕ⇒ ∆

Γ,¬¬ϕ⇒ ∆

Theorem

T `GCminψ iff T `M3

Lψ iff T `M3

Proof system for M3L and M3

Let the Gentzen-style system GCmin be obtained by replacingthe rule (¬ ⇒) in GCPL with the rule:

Γ, ϕ⇒ ∆

Γ,¬¬ϕ⇒ ∆

Theorem

T `GCminψ iff T `M3

Lψ iff T `M3

The logic Cmin

As the negation rules

Γ, ϕ⇒ ∆

Γ,¬¬ϕ⇒ ∆

Γ, ϕ⇒ ∆

Γ,⇒ ∆,¬ϕ

of GCmin translate to (¬¬ϕ ⊃ ϕ) and (¬ϕ ∨ ϕ), respectively,GCmin is equivalent to the system HCmin obtained by addingthe above axiom schemes to HCL+.

The system HCmin represents the paraconsistent logic Cmin

(studied by Carnielli and Marcos).

Expressive power of Nmatrices

M - a two-valued Nmatrix with at least one propernon-deterministic operation.

There is no finite family of finite ordinary matrices C , suchthat `M= `C .

If M includes the classical implication, then there is even nofinite family of finite ordinary matrices C such that `M ψ iff`C ψ.

Analyticity, decidability and compactness

An obvious, yet crucial fact: any partial M-legal valuationdefined on a set of formulas closed under subformulas can beextended to a full M-legal valuation.

If M is finite then this entails that `M is:

DecidableFinitary (the compactness theorem obtains)

Analyticity, decidability and compactness

An obvious, yet crucial fact: any partial M-legal valuationdefined on a set of formulas closed under subformulas can beextended to a full M-legal valuation.

If M is finite then this entails that `M is:

DecidableFinitary (the compactness theorem obtains)

Refinements

Let M1 = 〈V1,D1,O1〉 and M2 = 〈V2,D2,O2〉 be Nmatrices forL.

M1 is a refinement of M2 if:

1 V1 ⊆ V2

2 D1 = D2 ∩ V1

3 �M1 (x1 . . . xn) ⊆ �M2 (x1 . . . xn) for every n-ary connective � ofL and every x1 . . . xn ∈ V1.

If M1 is a refinement of M2 then `M2⊆ `M1 .

Examples of refinements

M3S is a refinement of M3

The classical two-valued matrix is a refinement of M3S .

The classical two-valued matrix is also a refinement of M2.

¬t {f}I Vf {t}

¬t {f}I Df {t}

¬t {t, f}f {t}

Canonical Gentzen-type systems - outline

Canonical Gentzen-type systems: abstract systems with all thestandard structural rules and logical rules of a natural form.

Two-valued Nmatrices allow to provide for these systems:

simple semanticsconstructive semantic characterization of proof-theoreticalproperties: cut-elimination, invertibility of rules,axiom-expansion.

What is a canonical rule?

An “ideal” logical rule: an introduction rule for exactly oneconnective, on exactly one side of a sequent.

In its formulation: exactly one occurrence of the introducedconnective, no other occurrences of other connectives.

The rule should also be pure (i.e. context-independent): noside conditions limiting its application.

Its active formulas: immediate subformulas of its principalformula.

What is a canonical rule?

Stage 1.Γ, ψ, ϕ⇒ ∆

Γ, ψ ∧ ϕ⇒ ∆

Γ⇒ ∆, ψ Γ⇒ ∆, ϕ

Γ⇒ ∆, ψ ∧ ϕ

Stage 2.ψ,ϕ⇒ψ ∧ ϕ⇒

⇒ ψ ⇒ ϕ

⇒ ψ ∧ ϕ

Stage 3.

{p1, p2 ⇒}/p1 ∧ p2 ⇒ {⇒ p1 ; ⇒ p2}/⇒ p1 ∧ p2

Example 1

Conjunction rules:

{p1, p2 ⇒} / p1 ∧ p2 ⇒ {⇒ p1 ; ⇒ p2} / ⇒ p1 ∧ p2

Their applications:

Γ, ψ, ϕ⇒ ∆

Γ, ψ ∧ ϕ⇒ ∆

Γ⇒ ∆, ψ Γ⇒ ∆, ϕ

Γ⇒ ∆, ψ ∧ ϕ

Example 2

Implication rules:

{p1 ⇒ p2} / ⇒ p1 ⊃ p2 {⇒ p1 ; p2 ⇒} / p1 ⊃ p2 ⇒

Their applications:

Γ, ψ ⇒ ∆, ϕ

Γ⇒ ∆, ψ ⊃ ϕΓ⇒ ∆, ψ Γ, ϕ⇒ ∆

Γ, ψ ⊃ ϕ⇒ ∆

Example 3

Semi-implication rules:

{⇒ p1 ; p2 ⇒} / p1 p2 ⇒ {⇒ p2} / ⇒ p1 p2

Their applications:

Γ⇒ ∆, ψ Γ, ϕ⇒ ∆

Γ, ψ ϕ⇒ ∆

Γ⇒ ∆, ϕ

Γ⇒ ∆, ψ ϕ

Example 4

“Tonk” rules:

{p2 ⇒} / p1Tp2 ⇒ {⇒ p1} / ⇒ p1Tp2

Their applications:

Γ, ψ ⇒ ∆

Γ, ϕTψ ⇒ ∆

Γ⇒ ∆, ϕ

Γ⇒ ∆, ϕTψ

Canonical systems

Reminder: a standard Gentzen-type system for L:

Γ⇒ ∆Γ, Γ′ ⇒ ∆,∆′

Γ⇒ ∆(Cut)

A standard Gentzen-type system is canonical if each of itslogical (i.e. non-structural) rules is canonical.

Examples: GCPL, GCLuN (but not GCmin)

Canonical systems

Reminder: a standard Gentzen-type system for L:

Γ⇒ ∆Γ, Γ′ ⇒ ∆,∆′

Γ⇒ ∆(Cut)

A standard Gentzen-type system is canonical if each of itslogical (i.e. non-structural) rules is canonical.

Examples: GCPL, GCLuN (but not GCmin)

What sets of rules are acceptable?

If G is a canonical system, then `G is a structural and finitarycr.

But is it a logic? i.e., is it also consistent?

What sets of rules are acceptable?

If G is a canonical system, then `G is a structural and finitarycr.

But is it a logic? i.e., is it also consistent?

Coherence

A canonical calculus G is coherent if for every pair of rulesΘ1/⇒ �(p1, ..., pn) and Θ2/ � (p1, ..., pn)⇒, the set ofclauses Θ1∪Θ2 is classically unsatisfiable (and so inconsistent,i.e., the empty sequent can be derived from it using only cuts)

For a canonical calculus G , `G is a logic iff G is coherent.

Coherent calculi:

{p1, p2 ⇒} / p1 ∧ p2 ⇒ {⇒ p1 ; ⇒ p2} / ⇒ p1 ∧ p2

{p1 ⇒ p2} / ⇒ p1 ⊃ p2 {⇒ p1 ; p2 ⇒} / p1 ⊃ p2 ⇒

{⇒ p1 ; p2 ⇒} / p1 p2 ⇒ {⇒ p2} / ⇒ p1 p2

{p1 ⇒} / ⇒ ¬p1 {⇒ p1} / ¬p1 ⇒

Non-coherent: “Tonk”!

{p2 ⇒} / p1Tp2 ⇒ {⇒ p1} / ⇒ p1Tp2

From these rules, we can derive p ⇒ q for any p, q:

p ⇒ p q ⇒ q

p ⇒ pTq pTq ⇒ q

p ⇒ q

Every coherent calculus has a characteristic 2Nmatrix

G0 - the canonical calculus over {∧, } with no canonical ruleswhatsoever.

ψ ϕ ψ∧ϕt t {t,f}t f {t,f}f t {t,f}f f {t,f}

ψ ϕ ψ ϕt t {t,f}t f {t,f}f t {t,f}f f {t,f}

Add the rule {p1, p2 ⇒}/p1 ∧ p2 ⇒, which can be split in{p1 ⇒}/p1 ∧ p2 ⇒ and {p2 ⇒}/p1 ∧ p2 ⇒

ψ ϕ ψ∧ϕt t {t,f}t f {f}f t {f}f f {f}

Add the rule {⇒ p1;⇒ p2}/⇒ p1 ∧ p2

ψ ϕ ψ∧ϕt t {t}t f {f}f t {f}f f {f}

Add the rule {⇒ p2}/⇒ p1 p2

ψ ϕ ψ ϕt t {t}t f {t,f}f t {t}f f {t,f}

Add the rule {⇒ p1 ; p2 ⇒}/p1 p2 ⇒

ψ ϕ ψ ϕt t {t}t f {f}f t {t}f f {t,f}

Every 2Nmatrix has a corresponding coherent calculus

p1 p2 p1◦p2

t t {f}t f {f}f t {t,f}f f {t}

{⇒ p1 ; ⇒ p2} / p1 ◦ p2 ⇒

{⇒ p1 ; p2 ⇒} / p1 ◦ p2 ⇒

{p1 ⇒ ; p2 ⇒} / ⇒ p1 ◦ p2

Exact correspondence

Theorem: If G is a canonical calculus, then the followingstatements are equivalent:

1 `G is consistent (and so it is a logic).

2 G is coherent.

3 G has a characteristic 2Nmatrix.

4 G admits cut-elimination.

Every 2Nmatrix has a corresponding coherent calculus

p1 p2 p1◦p2

t t {f}t f {f}f t {t,f}f f {t}

{⇒ p1 ; ⇒ p2} / p1 ◦ p2 ⇒

{⇒ p1 ; p2 ⇒} / p1 ◦ p2 ⇒

{p1 ⇒ ; p2 ⇒} / ⇒ p1 ◦ p2

This is not the most efficient form - this calculus can be“normalized”.

Canonical calculi in normal form

A canonical calculus G is in normal form if each connective Ghas at most one left introduction rule and at most one rightintroduction rule.

Any canonical calculus can be transformed into an equivalentnormal form.

Example

Consider the following two rules for the binary connective X(representing XOR):

{⇒ p1 ; p2 ⇒} / ⇒ p1Xp2 {⇒ p2 ; p1 ⇒} / ⇒ p1Xp2

{⇒ p1, p2 ; p1 ⇒ p1 ; p2 ⇒ p2 ; p1, p2 ⇒} / ⇒ p1Xp2

{⇒ p1, p2 ; p1, p2 ⇒} / ⇒ p1Xp2

The XOR connective

p1 p2 p1Xp2

t t {f}t f {t}f t {t}f f {f}

{⇒ p1, p2 ; p1, p2 ⇒}/⇒ p1Xp2

{p1 ⇒ p2 ; p2 ⇒ p1}/ p1Xp2 ⇒

When are canonical rules invertible?

Invertibility of a rule R of G

R is invertible in G if for every application of R it holds thatwhenever its conclusion is provable in G , also each of its premisesis provable in G .

Example:

R1 = {p1, p2 ⇒}/p1 ∧ p2 ⇒ R2 = {⇒ p1 ; ⇒ p2}/⇒ p1 ∧ p2

Invertibility of R1:

Γ, ψ1 ∧ ψ2 ⇒ ∆

Γ, ψ1 ⇒ ∆, ψ1 Γ, ψ2 ⇒ ∆, ψ2

Γ, ψ1, ψ2 ⇒ ∆, ψ1 ∧ ψ2

Γ, ψ1, ψ2 ⇒ ∆

When are canonical rules invertible?

Calculus G1:

{p1, p2 ⇒}/p1 ∧ p2 ⇒ {⇒ p1 ; ⇒ p2}/⇒ p1 ∧ p2

An (equivalent) calculus G2:

{p1 ⇒}/p1 ∧ p2 ⇒ {p2 ⇒}/p1 ∧ p2 ⇒

{⇒ p1 ; ⇒ p2}/⇒ p1 ∧ p2

The first rules are NOT invertible: `G2 ψ1 ∧ ψ2 ⇒ ψ1, but6`G2ψ1 ⇒ ψ2.

The reason: G2 is not in normal form!

Axiom expansion

Axiom expansion in a calculus allows for a reduction of logicalaxioms to the atomic case.

An n-ary connective � admits axiom expansion in a calculus Gif whenever �(p1, ..., pn)⇒ �(p1, ..., pn) is provable in G , ithas a cut-free derivation in G from atomic axioms of the form{pi ⇒ pi}1≤i≤n.

A yet another correspondence

Theorem

Let G be a coherent canonical calculus in normal form. Thefollowing statements are equivalent:

1 The rules of G are invertible,

2 G has a characteristic two-valued deterministic matrix.

3 Every connective of L admits axiom expansion in G .

Canonical Gentzen-type systems - summary

2Nmatrices correspond to the class of canonical systemswhich are coherent and admit cut-elimination.

Deterministic 2Nmatrices correspond to the class of normalcanonical systems with invertible rules and axiom-expansion.

Modular semantics

Simple constructive characterization of syntactic properties

Modular semantics

Next part

Another application of the framework of Nmatrices for Logics ofFormal (In)consistency:

Modular non-deterministic semantics

Systematic construction of cut-free proof systems

Thank you for your attention!

Tutorial on Non-Deterministic Semantics Part I:...

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